Two rods of different materials having coefficient of thermal expansion `alpha_(1), alpha_(2)` and young's modulii `Y_(1) ,Y_(2)` respectively are fixed between two rigid massive walls. The rods are heated such that they undergo the same increase in temperature. There is no bending of rods. If `alpha_(1) :alpha_(2)=2 : 3`, the thermal stresses developed in the two rods are equal provided `Y_(1) : Y_(2)` is equal to

Options

A. `2 : 3`
B. `1 :1`
C. `3 : 2`
D. `4 : 9`

Answer

C

Explanation

Expansion in rod due to rise in temp. = compression in rod
For 1 st rod. Expansion in rod due to rise in temp. `Delta theta` will be, `Delta l_(1)=alpha_(1) l_(1) Delta theta`
Compression in rod, `Delta l_(1)=(-F)/(A)xx(l_(1))/(Y_(1))`
As, the length of the rod remains unchanged
`:. alpha_(1)l_(1)Delta theta=-(F)/(A)(l_(1))/(Y_(1))`
or `alpha_(1)Delta theta=-(F)/(AY_(1))` .....(i)
Similarly for section rod,
`alpha_(2)Delta theta=-(F)/(AY_(2))` .....(ii)
`:. (alpha_(1))/(alpha_(2))=(Y_(2))/(Y_(1))` or `(Y_(1))/(Y_(2))=(alpha_(2))/(alpha_(1)) =(3)/(2)`